On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Source abstract

Let Q C ( K , g ) \mathcal {QC}(K,g) be a family of K K -quasiconformal mappings of the open unit disk onto itself satisfying the PDE Δ w = g \Delta w =g , g ∈ C ( U ¯ ) g\in C(\overline {\mathbb {U}}) , w ( 0 ) = 0 w(0)=0 . It is proved that Q C ( K , g ) \mathcal {QC}(K,g) is a uniformly Lipschitz family. Moreover, if | g | ∞ |g|_\infty is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as K → 1 K \to 1 and | g | ∞ → 0 |g|_\infty \to 0 , so w ∈ Q C ( K , g ) w\in \mathcal {QC}(K,g) behaves almost like a rotation for sufficiently small K K and | g | ∞ |g|_\infty .